We have talked at length about derivatives abstractly. But how do we actually find derivative functions? We will outline this in this exercise and the next one.

The answer is through a series of “rules” that we’ll introduce in this exercise and the next. These rules are building blocks, and by combining rules, we can find and plot the derivatives of many common functions.

To start, we can think that derivatives are linear operators, fancy language that means:

ddxcf(x)=cf(x)\frac{d}{dx} c f(x) = c f'(x)

This means that we can pull constants out when calculating a derivative. For example, say we have the following function:

f(x)=4x2f(x) = 4x^{2}

We can define the derivative as the following:

ddx4x2=4ddxx2\frac{d}{dx}4x^{2} = 4\frac{d}{dx}x^{2}

We can also say that

ddx(f(x)+g(x))=ddxf(x)+ddxg(x)\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)

This means that the derivative of a sum is the sum of the derivatives. For example, say we have the following function:

f(x)=x2+x3f(x) = x^{2} + x^{3}

We can define the derivative as the following:

ddx(x2+x3)=ddxx2+ddxx3\frac{d}{dx}(x^{2}+x^{3}) = \frac{d}{dx}x^{2}+ \frac{d}{dx}x^{3}

When products get involved, derivatives can get a bit more complicated. For example, let’s say we have a function in the following form of two separate “parts”:

f(x)=u(x)v(x)f(x) = u(x)v(x)

We define the product rule as:

f(x)=u(x)v(x)+v(x)u(x)f'(x) =u(x)v'(x) + v(x)u'(x)

Mnemonically, we remember this as “first times derivative of second plus second times derivative of first.” For example, say we have the following equation that is the product of two :

f(x)=x2log(x)f(x) = x^2log(x)

We would use the product rule to find the derivative of f(x):

f(x)=x2ddx(log(x))+log(x)ddx(x2)f'(x) = x^2\frac{d}{dx}(log(x))+log(x)\frac{d}{dx}(x^2)

Finally, one last rule to take with us into the next exercise is that the derivative of a constant is equal to 0:

ddxc=0\frac{d}{dx}c = 0

This is because for a constant there is never any change in value, so the derivative will always equal zero. For example, we can say:

ddx5=0\frac{d}{dx}5 = 0


Let’s use the applet to the right to get a more visual feel for derivatives. Use to slider to analyze the behavior of the following functions:

f(x)=x2f(x)=x\begin{aligned} f(x) = x^2 \\ f(x) = x \end{aligned}

By default you will see f(x)=x2 but you change between both function. Use the slider to see the tangent line on various points of the graph. Some things to take note of are:

  • For f(x)=x2, the tangent line has a negative slope for negative x values and a positive slope for positive x values.
  • For f(x)=x, the tangent line is always the same and on top of the function. This is because the slope is always the same (equal to one) for f(x)=x.

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